University of Bahrain
College of Science
Mathematics department
First Semester 2008-2009
Final Examination
Math 312
Duration: 2 hours
Date: 22 / 01 / 2009
Max. Mark: 50
Name:
ID Number:
Section:
Instructions:
1) Please check that this test has 6 questions and 8 pages.
2) Write your name, student number, and section in the above box.
Marking Scheme
Questions
Max. Mark
Mark. Obtained
1
8
2
8
3
8
4
8
5
8
6
10
Total
50
Good Luck
1
Question 1: [4 + 4 marks]
a) Let R be an integral domain with identity. Prove that if p is irreducible and u is a unit,
then pu is irreducible.
b) Let f(x) = x4 + n x3 + x2 + n  Z 7[x]. Find n if the polynomial g(x) = x - 2 divides f(x).
2
Question 2: [4 + 4 marks]
a) Find all irreducible polynomials of degree 2 in the polynomial rings
b) Show that, if a + bi is prime in Z [i], then a – bi is prime in Z [i].
3
Z 2[x].
Question 3: [4 + 4 marks]
a) Let R be an Euclidean domain with degree function . Prove that if (1)  -2, then the
function ’: R –{0}  N defined by ’(x) = (x) + 2 is also a degree function.
2
b) Let P = X + 2 X + 2 be a polynomial of
find its elements.
4
Z 3[X]. Show that Z 3[X]/(P)
is a field and
Question 4 [4 + 4 marks]
a) Let R be division ring. Prove that the center Z(R) of R is a field, where Z(R) is defined
by Z(R) = {a  R : a x = x a for all x  R}.
b) Let R be a Boolean ring and P be a prime ideal of R. Show that R/P has only two
elements (use the fact that x2 = x for all x  R). Then conclude that P is a maximal ideal.
5
Question 5 [4 + 4 marks]
Let R be a principal ideal domain and P be a prime ideal of R.
a) Prove that P is generated by a prime element.
b) Prove that P is a maximal ideal.
6
Question 6 [3 + 3 + 4 marks]
x
Let R be the set of all upper triangular matrices R  { 
0
y
: x , y  Z }.
z 
a) Prove that R is a ring with identity.
0 a 
b) Prove that J  {
 : a
0
0


Z } is an ideal of R.
x
c) By considering the function f : R  ZZ defined by f ( 
0
is not a prime ideal of R.
7
y
) = (x , z), show that J
z 
8